Quantum spin systems at positive temperature

We develop a novel approach to phase transitions in quantum spin models based on a relation to their classical counterparts. Explicitly, we show that whenever chessboard estimates can be used to prove a phase transition in the classical model, the corresponding quantum model will have a similar phase transition, provided the inverse temperature β and the magnitude of the quantum spins S satisfy β √ S. From the quantum system we require that it is reflection positive and that it has a meaningful classical limit; the core technical estimate may be described as an extension of the BerezinLieb inequalities down to the level of matrix elements. The general theory is applied to prove phase transitions in various quantum spin systems with S 1. The most notable examples are the quantum orbital-compass model on Z2 and the quantum 120-degree model on Z3 which are shown to exhibit symmetry breaking at low-temperatures despite the infinite degeneracy of their (classical) ground state.